Finite Group Databasebeta

← #26109193

Subgroups — #26109193

Order 12032.

460 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243426C212yesmaximal, complemented
2#20241726C212yesmaximal
3#20241265C212yesmaximal, complemented
4C2×C47×#435C211yesmaximal, characteristic
5C2×#13093960—22no—
6#15689087C2214yes—
7#15688888—22no—
8#15688849—22no—
9#15688847—22no—
10C22×M16×C47C411yescharacteristic
11C23×C8×C47C2211yescharacteristic
12C2×C4×D4×C47C2211yescharacteristic
13C47×#435—22no—
14C47×#435—22no—
15C2×C4⋊C8×C47C411yescharacteristic
16C2×C22⋊C8×C47C411yescharacteristic
17C2×C22⋊C8×C47C2211yescharacteristic
18C2×#97×C47C411yescharacteristic
19C2×#10625329—24no—
20#13093960—44no—
21C22⋊(C47⋊C8)—24no—
22C22⋊(C47⋊C8)—24no—
23C22×D4×C47Q811yescharacteristic
24C23×C4×C47C2×C411yescharacteristic
25C23×C4×C47C2311yescharacteristic
26C2×M16×C47—21no—
27C2×M16×C47—21no—
28C2×M16×C47—42no—
29C22×C8×C47—42no—
30C22×C8×C47C2×C411yescharacteristic
31C22×C8×C47C2×C411yescharacteristic
32C22×C8×C47C2×C411yescharacteristic
33C22×C8×C47C2×C411yescharacteristic
34C22×C8×C47D412yes—
35C22×C8×C47D412yes—
36C4×D4×C47—22no—
37C4×D4×C47—22no—
38C2×C4⋊C4×C47D411yescharacteristic
39C2×C22⋊C4×C47D411yescharacteristic
40C2×C22⋊C4×C47Q811yescharacteristic
41C2×C42×C47C2×C411yescharacteristic
42C4⋊C8×C47—22no—
43C22⋊C8×C47—22no—
44C22⋊C8×C47—22no—
45#97×C47—21no—
46#97×C47—21no—
47C24×C47C4⋊C411yescharacteristic
48C24×C47C2×Q811yescharacteristic
49C2×D4×C47—22no—
50C2×D4×C47—42no—
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—41no—
54C22×C4×C47—41no—
55C22×C4×C47—41no—
56C22×C4×C47—41no—
57C22×C4×C47C4⋊C411yescharacteristic
58C22×C4×C47C4⋊C411yescharacteristic
59C22×C4×C47C4⋊C411yescharacteristic
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C2×D412yes—
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47#4112yes—
64M16×C47—42no—
65M16×C47—42no—
66C2×C8×C47—22no—
67C2×C8×C47—24no—
68C2×C8×C47—21no—
69C2×C8×C47—21no—
70C2×C8×C47—41no—
71C2×C8×C47—42no—
72C2×C8×C47—41no—
73C2×C8×C47—41no—
74C2×C8×C47C22⋊C412yes—
75C2×C8×C47C22⋊C411yescharacteristic
76C2×C8×C47C22⋊C411yescharacteristic
77C4⋊C4×C47—22no—
78C22⋊C4×C47—22no—
79C22⋊C4×C47—22no—
80C42×C47—21no—
81C42×C47—21no—
82#10625329—44no—
83#10625329—44no—
84C2×C47⋊C8—44no—
85C23×C47—22no—
86C23×C47—21no—
87C23×C47—22no—
88C23×C47—41no—
89C23×C47—41no—
90C23×C47—41no—
91C23×C47—41no—
92C23×C47C2×C4⋊C411yescharacteristic
93C23×C47C22⋊Q812yes—
94D4×C47—44no—
95C2×C4×C47—22no—
96C2×C4×C47—21no—
97C2×C4×C47—22no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
103C2×C4×C47—41no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47C2×C22⋊C411yescharacteristic
111C2×C4×C47#11711yescharacteristic
112C2×C4×C47#12112yes—
113C8×C47—22no—
114C8×C47—21no—
115C8×C47—41no—
116C8×C47—41no—
117C8×C47#9912yes—
118C47⋊C8—84no—
119#8613—471nomaximal, Sylow
120C22×C47—21no—
121C22×C47—21no—
122C22×C47—22no—
123C22×C47—22no—
124C22×C47—22no—
125C22×C47—44no—
126C22×C47—41no—
127C22×C47—41no—
128C22×C47—41no—
129C22×C47—82no—
130C22×C47#38011yescharacteristic
131C22×C47#43012yes—
132C4×C47—22no—
133C4×C47—21no—
134C4×C47—41no—
135C4×C47—41no—
136C4×C47—41no—
137C4×C47C2×#9911yescharacteristic
138C4×C47#41011yescharacteristic
139C2×#435D4711yescharacteristic
140#1810—472no—
141#1465—472no—
142#1293—472no—
143C2×C47—22no—
144C2×C47—42no—
145C2×C47—81no—
146C2×C47#179911yescharacteristic
147C2×C47#181111yescharacteristic
148C2×C47#181311yescharacteristic
149C22×M16Dic4711yescharacteristic
150C23×C8C2×D4711yescharacteristic
151C2×C4×D4C2×D4711yescharacteristic
152#435—22no—
153#435—22no—
154C2×#108—942no—
155C2×C4⋊C8Dic4711yescharacteristic
156#407—474no—
157C2×C22⋊C8Dic4711yescharacteristic
158C2×C22⋊C8C2×D4711yescharacteristic
159C2×#97Dic4711yescharacteristic
160#341—942no—
161#324—942no—
162#323—942no—
163C47#861311yescharacteristic, complemented, Sylow
164C22×D4Dic9411yescharacteristic
165C23×C4C2×Dic4711yescharacteristic
166C23×C4C22×D4711yescharacteristic
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—42no—
170C2×M16—944no—
171C22×C8—42no—
172C22×C8C2×Dic4711yescharacteristic
173C22×C8C2×Dic4711yescharacteristic
174C22×C8C2×Dic4711yescharacteristic
175C22×C8C2×Dic4711yescharacteristic
176C22×C8C22⋊D4712yes—
177C22×C8C22⋊D4712yes—
178C4×D4—22no—
179C4×D4—22no—
180C2×C4⋊C4D18811yescharacteristic
181C2×C22⋊C4Dic9411yescharacteristic
182C2×C22⋊C4D18811yescharacteristic
183C2×C42C2×Dic4711yescharacteristic
184#108—1884no—
185C4⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—944no—
189C22⋊C8—944no—
190#97—21no—
191#97—21no—
192C24C4⋊Dic4711yescharacteristic
193C24C2×Dic9411yescharacteristic
194C2×D4—22no—
195C2×D4—42no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4C4⋊Dic4711yescharacteristic
203C22×C4C4⋊Dic4711yescharacteristic
204C22×C4C4⋊Dic4711yescharacteristic
205C22×C4C2×D18811yescharacteristic
206C22×C4#1062535012yes—
207C22×C4C22×Dic4711yescharacteristic
208C22×C4C2×C22⋊D4712yes—
209M16—42no—
210M16—42no—
211M16—1884no—
212M16—1884no—
213C2×C8—24no—
214C2×C8—22no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—1884no—
222C2×C8C22⋊Dic4711yescharacteristic
223C2×C8C22⋊Dic4712yes—
224C2×C8C22⋊Dic4711yescharacteristic
225C4⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C42—21no—
229C42—21no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23C2×C4⋊Dic4711yescharacteristic
238C23#1309400312yes—
239D4—44no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4C2×C22⋊Dic4711yescharacteristic
256C2×C4#1309398311yescharacteristic
257C2×C4#1309399812yes—
258C8—22no—
259C8—21no—
260C8—41no—
261C8—41no—
262C8—3764no—
263C8#1309393912yes—
264C22—21no—
265C22—21no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—44no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—82no—
274C22#1568898311yescharacteristic
275C22#1568917212yes—
276C4—22no—
277C4—21no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4C2×#1309393911yescharacteristic
282C4#1568909711yescharacteristic
283C2—22no—
284C2—42no—
285C2—81no—
286C2#2024337611yescharacteristic
287C2#2024343011yescharacteristic
288C2#2024343811yescharacteristic
subgroupG/H
centerC22#15688983
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#435C2
socleC22×C47#380
prime radicalC22×D4×C47Q8
cosocleC23×C4×C47C23
hypercenterC2×#435D47
hypocenterC47#8613
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#435

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20243426C2semidirect
3. #20241265C2semidirect
163. C47#8613semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202434262
#202417262
#202412652
C2×C47×#4352
#861347